Geometric modeling of a closed flat contour with the application of rational cubic curves

Abstract

The article describes the method of geometric modeling of a closed planar contour using rational cubic curves. When modeling, the task was to make a contour from the minimum number of curve segments not more than third order. An approximation is made for the turbine blade profile. To do this, a flat closed contour of the blade is divided into two parts. The upper and lower parts of the approximating contour at the inflection points of the blade profile have common tangents. The upper part of the contour bounded by inflection points consists of two segments of rational third-order curves having the same curvature at a common point. The first segment of the cubic curve passes through three points of the given contour of the turbine blade, and at these points has tangents common with this contour. Moving the singular point of the first cubic curve within certain limits allows you to change its shape, as well as draw a segment of the curve through an additional point of the given contour and guarantees the absence of a singular point, break and inflection points within the of segment. After constructing the first segment of the cubic curve, the radius of curvature at the endpoint of the segment is determined. The segment of the second cubic curve, like the segment of the first curve, has three common points and tangents in them with a given contour of the turbine blade. Moving the singular point of the second cubic curve within the arc of the previously defined curve provides the specified curvature at the junction with the segment of the first cubic curve, and also guarantees the absence of a singular point, break points, and inflection points within the of segment. The lower part of the turbine blade profile is approximated by a segment of a second-order curve that passes through three points of the blade profile, and at the end points shares common tangents with the contour. The equations of the curves are determined in parametric form in the projective plane, and then written in the affine plane in vector-parametric form. The proposed method can be used both for modeling closed planar contours and for modeling planar contours of the second order of smoothness, the segments of which are rational curves of the third order.